The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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Prove $e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X}$ if $\hat{X}^{2}=I$.

If we have an operator $\hat{X}$ such that $\hat{X}^{2}=I$ (the identity), how do we prove that: $$e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X} \ ?$$
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1answer
32 views

positive elements of $C^*$-algebra

If $A$ is a abelian $C^*$-algebra and $a,b$ are elements in $A$ such that $0‎\leq ‎a‎\leq ‎1,0‎\leq ‎b‎\leq ‎1‎‎$ then $0‎\leq ‎a‎b\leq ‎1‎$. My problem is:" Is it true if $A$ is not abelian?"
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1answer
52 views

States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
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1answer
29 views

Why is a von Neumann algebra is closed with respect to weak * topology?

I was trying to prove that the identity map between a von Neumann algebra $(A,\mbox{ultra weak topology})$ with respect to ultra weak topology and the von Neumann algebra $A$ with respect to weak* ...
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1answer
40 views

Ultra weakly continuous linear functional on von neuman algebras

I started to learn von Neumann algebras and I wonder about the proof of the following. Can anyone outline it? this is theorem 4.2.10 in Murphy's book: A linear functional $\tau$ on von Neuman ...
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1answer
25 views

inner product and hermitian matrices

One of my professors mentioned that since a matrix A is positive semi definite and B is hermitian, hence the inner product $<A,B>$ is real. Is this an if and only if condition? So if we know ...
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1answer
77 views

If $0\leq a \leq b$ and $a$ is invertible, then $b$ is invertible

Let $\mathscr A$ be a unital C*-algebra and let $a,b\in \mathscr A$ such that $0\leq a \leq b$ and $a$ is invertible. How to show that $b$ is invertible? ($0\leq a \leq b$ means that $a,b$ is ...
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38 views

Unitary elements in C*-algebras

Let $\mathscr A$ be an unital $C^*$-algebra and let $u \in \mathscr A$ be an unitary element (I.e., $u^*u=uu^*=1$). Is it true that $$u=e^{ia}$$ for some hermitian element $a\in \mathscr A$? Im not ...
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22 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let ...
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1answer
49 views

Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
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38 views

A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
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22 views

Structure of crossed product $C(X)\rtimes_r G$ when $G$ is locally finite

Let $G$ be a discrete group acting on a compact Hausdorff space $X$. Then we may consider the reduced crossed product $C^*$-algebra $C(X)\rtimes_r G$. When $G$ is finite, I know that we essentially ...
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Operators problem

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having ...
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136 views

Measure theory , Functional calculus, Self Adoint

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
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1answer
75 views

Hilbert Space multiplication Operator, shift operator

I have this problem and am not sure how to even approach it.. Hilbert space $l^2(\mathbb{Z})$ with orthonormal basis$ $$(e_n)$ and Hamiltonian operator $He_n=i(e_{n+1}-e_{n-1})$ a)I need to use ...
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1answer
22 views

adjoint operator of the partial trace map

Could someone explain to me, what is the adjoint map of the partial trace map the (tensored with the identity map), or why does the following equality hold? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes ...
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$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as ...
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38 views

C*-Algebra generated by self-adjoint Elements of commutative unital C*-Algebra

I want to proof the following theorem: Let A be a commutative unital C*-Algebra and $a_1, ..., a_n\in A_{sa}$. Then $C^*(1_A, a_1, ..., a_n)\subseteq A$ is *-isomorphic to $C(\Omega )$ for a compact ...
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49 views

An strange operator in B(H)

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis for $H$. Let $E_0$ be a countable subset of $E$ and $\{\delta_i\}_1^{\infty}$ be a bounded sequence of $(0,\infty)$. For ...
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1answer
33 views

What is the definition of $\ell^2(G)$ where $G$ is a group?

First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have ...
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1answer
33 views

Is a crossed product of a separable $C^\ast$-algebra by a finite group separable?

If $A$ is a separable $C^\ast$-algebra, $\alpha$ is an action on $A$ by a finite group $G$, then is the crossed product $A\rtimes_\alpha G$ separable?
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35 views

$C^*$-seminorm smaller than C*-norm?

I am currently reading Ruy Exel's Partial Dynamical Systems, Fell Bundles and Applications where he mentions that for every $C^*$-seminorm $p$ on a C* algebra $A$ one has $$p(a)\leq ||a||$$ for all $a ...
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Construct a projection satisfying a certain property

Let $\cal G$ be a group of finite order $n$. For every prime divisor $p$ of $n$, construct a projection $P\in \cal N(G)$ such that $\operatorname{tr}_{\cal N(G)}(P)=1/p$. Here $\cal N(G)$ denotes ...
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1answer
31 views

Differential Operators and Coefficients

First question on Math StackExchange here. I have been staring at this for a bit, but wasn't quite able to get the hang of it. Here it goes. We are given \begin{align} \frac{\partial}{\partial x} = ...
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21 views

semifinite projections

Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$. ( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
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1answer
35 views

Maximal Ideals in $L^1(\mathbb{R})$

Define $I_\xi = \{ f \in L^1(\mathbb{R}) : \hat{f}(\xi)=0 \}$. I have to prove that $I_\xi$ is a maximal ideal in $L^1(\mathbb{R})$. The following are my attempts at solution : Attempt 1 : Consider ...
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1answer
35 views

finite and properly infinite projections

Let $M$ be a von Neumann algebras and $p$ be a projection in $M$. $Q1:$I want to prove that there is a central projection $z \in M$ such that $pz$ is finite and $P(1-z)$ is properly infinite. $Q2:$ ...
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prove that all pure states in a commutative C* algebra are multiplicative linear functionals

I am trying to prove this , but can not see it clearly. it was given as some sort of converse of the fact that all multiplicative linear functionals are pure states
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185 views

idempotents in a subalgebra of $B(H)$.

Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is closed under weak operator topology. Suppose that there exist idempotents ...
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1answer
26 views

$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
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Functoriality in $K$-theory for $C^*$-algebras or Banach algebras

I'm trying to clear up some confusion I'm having over how one establishes functoriality in $K$-theory for $C^*$-algebras or Banach algebras. Let me stick to $K_0$. Given a *-homomorphism (or bounded ...
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1answer
50 views

Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...
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1answer
35 views

Relative weak-star topology on pure states

Let $A$ be a (unital) C*-algebra and consider $PS(A)$, the set of all pure states on $A$ with the relative weak-star topology. I would like to check (a weaker form of) Uryshon's lemma on $PS(A)$ in ...
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1answer
38 views

Inverse Tensor map

Let $$\phi: M_n (\mathbb{C})\otimes M_m (\mathbb{C})\to M_{nm} (\mathbb{C})$$ $$ \phi({A}\otimes{B}) = \begin{bmatrix} a_{11} {B} & \cdots & a_{1n}{B} \\ \vdots & \ddots & \vdots \\ ...
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1answer
36 views

Holomorphic Functional Calculus for the Square Root

I'm working on a problem set, so I'm not looking for a solution, but just maybe a pointer on where I'm going wrong. I want to use the holomorphic functional calculus to determine the square root of ...
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0answers
23 views

A formula for representation

Let $A$ be a C*-algebra. Do you confirm the following discussion? Let us consider a representation $\pi:A\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$. If we denote ...
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Approximation by elements in intersection of two Banach subalgebras

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there ...
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1answer
30 views

Minimal projections II

Let $M_1$ and $M_2$ be two W*-algebras. Let $A$ be a C*-algebra and $\pi_j:A\to M_j$ be two faithful representations with $M_j=\overline{\pi_j(A)}^{w^*}$. Assume that $$\textrm{The unit of}~ ...
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1answer
23 views

Minimal projections

Assume $M$ is a W*-algebra such that the set of minimal projections is not empty. Let $z(M)$ be the supremum of all minimal projections in $M$. It is well-known that $z(M)$ is a central projection. ...
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20 views

Projections in *-homomorphism

Let us consider the commutative C*-algebra $C_0(\Omega)$ and a representation $\pi:C_0(\Omega)\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(C_0(\Omega))$. It is ...
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Operator system of minimal dimension with one dimensional projections

Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in ...
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Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
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1answer
26 views

Norm on reduced crossed product - $C^*$ version v.s. $L^p$ version

Let $(G,A,\alpha)$ be a $C^*$-dynamical system where $G$ is a countable discrete group. When defining the reduced crossed product, one can proceed as follows: Let $\pi$ be a faithful representation ...
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1answer
30 views

projection in a factor von Neumann algebra.

We know that center of a factor von Neumann algebra $\mathcal{A} $ is trivial. Let $P_1$ be a projection in $\mathcal{A} $ such that $P_1\neq I,0$ . undoubtedly there exist another projection like ...
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2answers
32 views

$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras.

Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$. I read in Murphy 2.1.11 that this is true if $B$ is ...
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43 views

Self-adjoint and positive operator minimal polynomial on complex inner product spaces

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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Positive operator minimal polynomial [duplicate]

Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers ...
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2answers
99 views

Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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30 views

An strange trace class operator

Assume $H$ is a non separable Hilbert space. 1- Let $\{a_n\}$ be a sequence of bounded linear operators on $H$. Any operator $a_n$ induces the bounded linear functional $x\to tr(xa_n)$ on $L^1(H)$, ...
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1answer
41 views

Showing that there exists a functional on $l^\infty$ whose value lies between the infimum and supremum of a sequence-is my solution correct?

I'm trying to show that there exists a functional in $l^\infty$ whose value lies between the infimum and supremum of a sequence. That is, there exists a functional $\phi:l^\infty\longrightarrow ...